For this functionality to achieve you need to use RobotActivity from Toolbox and automation will automatically start picking the Task from Pega Robotic Designer Studio. There is no use of clicking on any button in Pega Designer Studio.
All the machines which will run the same Runtime, Pega Designer studio will assign the cases to all machine and will keep a track of them automatically.
Hi, KaranG62 - depending on your use case you may find this post useful. It deals with triggering an automation using a button in Pega application. But it really depends on what kind of automation you are trying to accomplish.